If $a$,$b$,and $c$ are non-zero real numbers,then the minimum value of the expression $\left( \frac{(a^4 + a^2 + 1)(b^4 + 7b^2 + 1)(c^4 + 11c^2 + 1)}{a^2 b^2 c^2} \right)$ is

The maximum value of $3\cos \theta - 4\sin \theta$ is

If $P = \frac{1}{2} \sin^2 \theta + \frac{1}{3} \cos^2 \theta$, then:

The maximum value of $\cos \alpha_1 \cdot \cos \alpha_2 \cdots \cos \alpha_n$ under the restrictions $0 \le \alpha_1, \alpha_2, \dots, \alpha_n \le \frac{\pi}{2}$ and $\cot \alpha_1 \cdot \cot \alpha_2 \cdots \cot \alpha_n = 1$ is

If $\left| a \sin^2 \theta + b \sin \theta \cos \theta + c \cos^2 \theta - \frac{1}{2}(a + c) \right| \le \frac{1}{2}k,$ then $k^2$ is equal to

The maximum value of $(7 \cos\theta + 24 \sin\theta) \times (7 \sin\theta - 24 \cos\theta)$ for every $\theta \in R$.